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revised Is the empty set internal?
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revised Which universities teach true infinitesimal calculus?
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answered If $f,g$ are uniformly continuous prove $f+g,fg$ are uniformly continuous
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comment Understanding infinity
Oh yes you can.
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revised Have any definitions in mathematics been redefined
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comment Have any definitions in mathematics been redefined
@TobiasKildetoft, Thanks, good point.
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revised Have any definitions in mathematics been redefined
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answered Have any definitions in mathematics been redefined
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comment Lagrange's original proof of Remainder Theorem?
Which historical works have you consulted?
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revised Are the real numbers really uncountable?
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comment Are the real numbers really uncountable?
... these numbers don't "exist" in a constructive setting. From the constructive viewpoint, their dubious "existence" is wholly dependent on an unbridled application of the law of excluded middle, rejected by constructivists (again with the proviso that Cantor's diagonalisation argument remains meaningful in a constructive setting as well; see Bishop's book).
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comment Are the real numbers really uncountable?
@IncnisMrsi, the various characterisations of the reals are going to be equivalent as you said but again only in the context of classical logic. In constructive mathematics the least upper bound property fails; see for instance Bishop's book, page 4. When classical logic is the background logic, what is responsible for the uncountability of the reals is the "presence" of undefinable real numbers. The OP was obviously puzzled by this: how can a number exist if you can't specify it at all? What I was pointing out is that, in fact,...
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revised Which universities teach true infinitesimal calculus?
as per Thomas's suggestion
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awarded  Nice Question
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comment Are the real numbers really uncountable?
@IncnisMrsi, it is hard to discuss a mathematical concept if we don't agree on its definition. What is then your basic concept of real numbers? As far as the least upper bound is concerned you could consult en.wikipedia.org/wiki/Least-upper-bound_property
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revised Is the empty set internal?
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revised Is the outer measure of $A\cup B$ equal to the sum of their outer measures if $A\cap B=\varnothing$?
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awarded  Popular Question
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comment Is $\frac{\textrm{d}y}{\textrm{d}x}$ not a ratio?
Over the hyperreals, $\frac{dy}{dx}$ is a ratio and one can view $\frac{d}{dx}$ as an operator. Therefore Tobin's reply is not a good argument for "telling that dy/dx is not a fraction".
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revised Are the real numbers really uncountable?
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